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Monte-Carlo methods are widely used numerical tools in various fields of application, like rarefied gas dynamics, vacuum technology, stellar dynamics or nuclear physics. A central part in all applications is the generation of random variates according to a given probability law. Fundamental techniques to generate non-uniform random variates are the inversion principle or the acceptance-rejection method. Both procedures can be quite time-consuming if the given probability law has a complicated structure.; In this paper we consider probability laws depending on a small parameter and investigate the use of asmptotic expansions to generate random variates. The results given in the paper are restrictedto first order expansions. We show error estimates for the discrepancy as well as for the bounded Lipschitz distance of the asymptotic expansion. Furthermore the integration error for some special classes of functions is given. The efficiency of the method is proved by a numerical example from rarefied gas flows.

Particle methods to simulate rarefied gas flows have found an increasing interest in Computational Fluid Dynamics during the last decade, see for example [1], [2], [3] and [4]. The general goal is to develop numerical schemes which are reliable enough to substitute real windtunnel experiments, needed for example in space research, by computer experiments. In order to achieve this goal one needs numerical methods solving the Boltzmann equation including all important physical effects. In general this means 3D computations for a chemically reacting rarefied gas. With codes of this kind at hand, Boltzmann simulation becomes a powerful tool in studying rarefied gas phenomena.

The paper presents some approximation methods for the Boltzmann equation. In the first part fully implicit discretization techniques for the spatially homogeneous Boltzmann equation are investigated. The implicit equation is solved using an iteration process. It is shown that the iteration converges to the correct solution for the moments of the distribution function as long as the mass conservation is strictly fulfilled. For a simple model Boltzmann equation some unexpected features of the implicit scheme and the corresponding iteration process are clarified. In the second part a new iteration algorithm is proposed which should be used for the stationary Boltzmann equation. The realization of the method is very similar to the standard splitting algorithms except some new stochastic elements.